Kern- und Teilchenphysik II Lecture 3: Weak Interaction

(adapted from the Handout of Prof. Mark Thomson) Prof. Nico Serra Dr. Marcin Chrzaszcz Dr. Annapaola De Cosa (guest lecturer)

www.physik.uzh.ch/de/lehre/PHY213/FS2017.html Parity

« The parity operator performs spatial inversion through the origin:

• applying twice: so • To preserve the normalisation of the wave-function

Unitary • But since Hermitian which implies Parity is an observable quantity. If the interaction Hamiltonian commutes with , parity is an observable conserved quantity • If is an eigenfunction of the parity operator with eigenvalue

since Parity has eigenvalues « QED and QCD are invariant under parity « Experimentally observe that Weak Interactions do not conserve parity

Mark Thomson/Nico Serra 2 Nuclear and Particle Physics II Intrinsic Parity

Intrinsic Parities of fundamental particles: Spin-1 Bosons • From Gauge Field Theory can show that the gauge bosons have

Spin-½ Fermions • From the Dirac equation showed (see Dirac Equation Lectures in KTI): Spin ½ particles have opposite parity to spin ½ anti-particles • Conventional choice: spin ½ particles have

and anti-particles have opposite parity, i.e.

« For Dirac spinors it was shown that the parity operator is:

Mark Thomson/Nico Serra 3 Nuclear and Particle Physics II Parity conservation

– – • Consider the QED process e q ¦ e q e– e– • The Feynman rules for QED give:

• Which can be expressed in terms of the electron and quark 4-vector currents: q q

with and « Consider the what happen to the matrix element under the parity transformation s Spinors transform as

s Adjoint spinors transform as

s Hence

Mark Thomson/Nico Serra 4 Nuclear and Particle Physics II Parity Conservation

« Consider the components of the four-vector current 0: since

k=1,2,3: since • The time-like component remains unchanged and the space-like components change sign • Similarly or « Consequently the four-vector scalar product

QED Matrix Elements are Parity Invariant

Parity Conserved in QED

« The QCD vertex has the same form and, thus, Parity Conserved in QCD

Mark Thomson/Nico Serra 5 Nuclear and Particle Physics II Parity Violation in Beta decays

« The parity operator corresponds to a discrete transformation « Under the parity transformation: Vectors change sign Note B is an axial vector Axial-Vectors unchanged « 1957: C.S.Wu et al. studied beta decay of polarized cobalt-60 nuclei:

« Observed electrons emitted preferentially in direction opposite to applied field

If parity were conserved: expect equal rate for producing e– in directions along and opposite to the nuclear spin. more e- in c.f. « Conclude parity is violated in WEAK INTERACTION that the WEAK interaction vertex is NOT of the form

Mark Thomson/Nico Serra 6 Nuclear and Particle Physics II Bilinear Covariants

« The requirement of Lorentz invariance of the matrix element severely restricts the form of the interaction vertex. QED and QCD are VECTORinteractions:

« This combination transforms as a 4-vector (Handout 2 appendix V) « In general, there are only 5 possible combinations of two spinors and the gamma matrices that form Lorentz invariant currents, called bilinear covariants: Type Form Components Boson Spin s SCALAR 1 0 s PSEUDOSCALAR 1 0 s VECTOR 4 1 s AXIAL VECTOR 4 1 s TENSOR 6 2 « Note that in total the sixteen components correspond to the 16 elements of a general 4x4 matrix: decomposition into Lorentz invariant combinations « In QED the factor arose from the sum over polarization states of the virtual photon (2 transverse + 1 longitudinal, 1 scalar) = (2J+1) + 1 « Associate SCALAR and PSEUDOSCALAR interactions with the exchange of a SPIN-0 boson, etc. – no spin degrees of freedom

Mark Thomson/Nico Serra 7 Nuclear and Particle Physics II V-A in Weak interactions

« The most general form for the interaction between a fermion and a boson is a linear combination of bilinear covariants « For an interaction corresponding to the exchange of a spin-1 particle the most general form is a linear combination of VECTOR and AXIAL-VECTOR « The form for WEAK interaction is determined from experiment to be VECTOR – AXIAL-VECTOR (V – A) e– νe

V – A

« Can this account for parity violation? « First consider parity transformation of a pure AXIAL-VECTOR current with

or

Mark Thomson/Nico Serra 8 Nuclear and Particle Physics II V-A in Weak interactions

• The space-like components remain unchanged and the time-like components change sign (the opposite to the parity properties of a vector-current)

• Now consider the matrix elements

• For the combination of a two axial-vector currents

• Consequently parity is conserved for both a pure vector and pure axial-vector interactions • However the combination of a vector current and an axial vector current

changes sign under parity – can give parity violation !

Mark Thomson/Nico Serra 9 Nuclear and Particle Physics II V-A in Weak interactions

« Now consider a general linear combination of VECTOR and AXIAL-VECTOR (note this is relevant for the Z-boson vertex)

• Consider the parity transformation of this scalar product

• If either gA or gV is zero, Parity is conserved, i.e. parity conserved in a pure VECTOR or pure AXIAL-VECTOR interaction

• Relative strength of parity violating part

Maximal Parity Violation for V-A (or V+A)

Mark Thomson/Nico Serra 10 Nuclear and Particle Physics II Chiral Structure

« Recall (QED lectures in KTI) introduced CHIRAL projections operators

project out chiral right- and left- handed states « In the ultra-relativistic limit, chiral states correspond to helicity states « Any spinor can be expressed as:

• The QED vertex in terms of chiral states:

conserves chirality, e.g.

« In the ultra-relativistic limit only two helicity combinations are non-zero

Mark Thomson/Nico Serra 11 Nuclear and Particle Physics II Helicity Structure in Weak Interaction

– «The charged current (W±) weak vertex is: e νe

« Since projects out left-handed chiral particle states: (question 16) « Writing and from discussion of QED, gives

Only the left-handed chiral components of particle spinors and right-handed chiral components of anti-particle spinors participate in charged current weak interactions « At very high energy , the left-handed chiral components are helicity eigenstates : LEFT-HANDED PARTICLES Helicity = -1

RIGHT-HANDED ANTI-PARTICLES Helicity = +1

Mark Thomson/Nico Serra 12 Nuclear and Particle Physics II Helicity Structure in Weak Interaction

In the ultra-relativistic limit only left-handed particles and right-handed antiparticles participate in charged current weak interactions

e.g. In the relativistic limit, the only possible electron – neutrino interactions are: e– e+ νe νe νe

e– « The helicity dependence of the weak interaction parity violation e.g.

RH anti-particle LH particle RH particle LH anti-particle

Valid weak interaction Does not occur

Mark Thomson/Nico Serra 13 Nuclear and Particle Physics II Helicity in pion decay

« The decays of charged pions provide a good demonstration of the role of helicity in the weak interaction

EXPERIMENTALLY:

• Might expect the decay to electrons to dominate – due to increased phase space…. The opposite happens, the electron decay is helicity suppressed « Consider decay in pion rest frame. • Pion is spin zero: so the spins of the ν and µ are opposite • Weak interaction only couples to RH chiral anti-particle states. Since neutrinos are (almost) massless, must be in RH Helicity state • Therefore, to conserve angular mom. muon is emitted in a RH HELICITY state

• But only left-handed CHIRAL particle states participate in weak interaction

Mark Thomson/Nico Serra 14 Nuclear and Particle Physics II Helicity in pion decay

« The general right-handed helicity solution to the Dirac equation is

with and

• project out the left-handed chiral part of the wave-function using

giving

In the limit this tends to zero

• similarly

In the limit ,

Mark Thomson/Nico Serra 15 Nuclear and Particle Physics II Helicity in pion decay

« Hence

RH Helicity RH Chiral LH Chiral

• In the limit , as expected, the RH chiral and helicity states are identical • Although only LH chiral particles participate in the weak interaction the contribution from RH Helicity states is not necessarily zero !

mν ≈ 0: RH Helicity ≡ RH Chiral mµ ≠ 0: RH Helicity has LH Chiral Component

« Expect matrix element to be proportional to LH chiral component of RH Helicity electron/muon spinor from the kinematics of pion decay at rest

« Hence because the electron mass is much smaller than the pion mass the decay is heavily suppressed.

Mark Thomson/Nico Serra 16 Nuclear and Particle Physics II Evidence of V-A

« The V-A nature of the charged current weak interaction vertex fits with experiment EXAMPLE charged pion decay (question 17) • Experimentally measure:

• Theoretical predictions (depend on Lorentz Structure of the interaction)

V-A or V+A

Scalar or Pseudo-Scalar

EXAMPLE muon decay Measure electron energy and angular distributions relative to muon spin direction. Results expressed in terms of general S+P+V+A+T form in Michel Parameters

e.g. TWIST expt: 6x109 µ decays Phys. Rev. Lett. 95 (2005) 101805 V-A Prediction:

Mark Thomson/Nico Serra 17 Nuclear and Particle Physics II W Propagator

« The charged-current Weak interaction is different from QED and QCD in that it is mediated by massive W-bosons (80.3 GeV) « This results in a more complicated form for the propagator: • in handout 4 showed that for the exchange of a massive particle: massless massive

• In addition the sum over W boson polarization states modifies the numerator W-boson propagator spin 1 W±

« However in the limit where is small compared with the interaction takes a simpler form. W-boson propagator ( )

• The interaction appears point-like (i.e no q2 dependence)

Mark Thomson/Nico Serra 18 Nuclear and Particle Physics II Connection to Fermi Theory

« In 1934, before the discovery of parity violation, Fermi proposed, in analogy with QED, that the invariant matrix element for β-decay was of the form:

where • Note the absence of a propagator : i.e. this represents an interaction at a point « After the discovery of parity violation in 1957 this was modified to

(the factor of √2 was included so the numerical value of GF did not need to be changed) « Compare to the prediction for W-boson exchange

which for becomes:

Still usually use to express strength of weak interaction as the is the quantity that is precisely determined in muon decay

Mark Thomson/Nico Serra 19 Nuclear and Particle Physics II Strength of Weak Interaction

« Strength of weak interaction most precisely measured in muon decay • Here • To a very good approximation the W-boson propagator can be written

• In muon decay measure • Muon decay

« To obtain the intrinsic strength of weak interaction need to know mass of W-boson: (see handout 14)

The intrinsic strength of the weak interaction is similar to, but greater than, the EM interaction ! It is the massive W-boson in the propagator which makes it appear weak. For weak interactions are more likely than EM.

Mark Thomson/Nico Serra 20 Nuclear and Particle Physics II